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A local energy estimate for 2-dimensional Dirichlet wave equations

Kellan Hepditch and Jason Metcalfe

Vol. 16 (2023), No. 3, 483–492
Abstract

We examine a variant of the integrated local energy estimate for (1+2)-dimensional Dirichlet wave equations exterior to star-shaped obstacles. The classical bound on the solution, rather than the derivative, is not typically available in two spatial dimensions. Using an argument inspired by the rp-weighted method of Dafermos and Rodnianski and taking advantage of the Dirichlet boundary conditions allow for the recovery of such a term when the initial energy is appropriately weighted.

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Keywords
wave equation, local energy estimate, exterior domain
Mathematical Subject Classification
Primary: 35L71, 35L05
Milestones
Received: 7 February 2022
Revised: 15 June 2022
Accepted: 11 September 2022
Published: 10 August 2023

Communicated by Suzanne Lenhart
Authors
Kellan Hepditch
Cedar Ridge High School
Hillsborough, NC
United States
Jason Metcalfe
Department of Mathematics
University of North Carolina
Chapel Hill, NC
United States